11.1 The 3 + 1 approaches

There is a lot of literature on the canonical formulation of general relativity both in the traditional
ADM and the Møller tetrad (or, recently, the closely related complex Ashtekar) variables. Thus it is quite
surprising how little effort has been spent to systematically quasi-localize them. One motivation for the
quasi-localization of the ADM-Regge-Teitelboim analysis came from the need for the microscopic
understanding of black hole entropy [32, 31, 99]: What are the microscopic degrees of freedom behind the
phenomenological notion of black hole entropy? Since the aim of the present paper is to review the
construction of the quasi-local quantities in classical general relativity, we discuss only the classical 2-surface
observables by means of which the ‘quantum edge states’ on the black hole event horizons were intended to
be constructed.

11.1.1 The 2-surface observables

If , the 3-manifold on which the ADM canonical variables , are defined, has a boundary
, then the usual vacuum constraints

are differentiable with respect to the canonical variables only if the smearing fields and and the derivative are
vanishing on 21.
However, as Balachandran, Chandar, and Momen [32, 31] and Carlip [99] realized, the boundary
conditions for the smearing fields can be relaxed by adding appropriate boundary terms to the constraints.
Namely, for any vector field and function they define

where is the trace of the extrinsic curvature of in and is the induced metric on .
Then is functionally differentiable if is tangent to , and is functionally
differentiable if is vanishing on (and hence, in particular, is a [not necessarily
zero] constant on ) and is fixed on . Furthermore, for any two such and
coinciding on , the difference is just a momentum constraint,
and, similarly, is a Hamiltonian constraint. Thus, on the constraint surface,
and depend only on the value of and on . A direct calculation
shows that their Poisson bracket with the constraints is a constraint, i.e. vanishing on the
constraint surface. Therefore, andare well-defined 2-surface observablescorresponding to the momentum and the Hamiltonian constraint, respectively. Moreover, the
observables form an infinite-dimensional Lie algebra with respect to the Poisson
bracket. In this Lie algebra the momentum constraints form an ideal, and the quotient of the
algebra of the observables and the constraint ideal is still infinite-dimensional. (In
the asymptotically flat case, cf. [47, 364].) In fact, defines a homomorphism of the Lie
algebra of the 2-surface vector fields into the quotient algebra of these observables modulo
constraints [32, 31, 99].
To understand the meaning of these observables, recall that any vector field on generates a
diffeomorphism, which is an exact (gauge) symmetry of general relativity, and the role of the momentum
constraint is just to generate this gauge symmetry in the phase space. However, the
boundary breaks the diffeomorphism invariance of the system, and hence on the boundary the
diffeomorphism gauge motions yield the observables and the gauge degrees of freedom
give raise to physical degrees of freedom, making it possible to introduce the so-called edge
states [32, 31, 99]. Evaluating and on the constraint surface we obtain

i.e. these are just the integrals of the (unreferenced) Brown-York energy surface density weighted
by the constant and momentum surface density contracted with , respectively (see
Equation (74)).

Analogous investigations were done by Husain and Major in [210]. Using Ashtekar’s complex
variables [17] they determined all the local boundary conditions for the canonical variables , and
for the lapse , the shift , and the internal gauge generator on that ensure the
functional differentiability of the Gauss, the diffeomorphism, and the Hamiltonian constraints.
Although there are several possibilities, they discussed the two most significant cases. In the first
case the generators , , and are vanishing on , whenever there are infinitely
many 2-surface observables both from the diffeomorphism and the Gauss constraints, but no
observable from the Hamiltonian constraint. The structure of these observables is similar to that
of those coming from the ADM diffeomorphism constraint above. The other case considered
is when the canonical momentum (and hence, in particular, the 3-metric) is fixed on
the 2-boundary. Then the quasi-local energy could be an observable, as in the ADM analysis
above.

All of the papers [32, 31, 99, 210] discuss the analogous phenomenon of how the gauge freedoms are
getting to be true physical degrees of freedom in the presence of 2-surfaces on the 2-surfaces themselves in
the Chern-Simons and BF theories. Weakening the boundary conditions further (allowing certain boundary
terms in the variation of the constraints) a more general algebra of ‘observables’ can be obtained [101, 296]:
They form the Virasoro algebra with a central charge. (In fact, Carlip’s analysis in [101] is based on the
so-called covariant Noether charge formalism below.) Since this algebra is well known in conformal field
theories, this approach might be a basis of understanding the microscopic origin of the black hole
entropy [100, 101, 102, 296, 103]. However, this quantum issue is beyond the scope of the present
review.

"Quasi-Local Energy-Momentum and Angular Momentum in
GR: A Review Article"